rtsiandle optimalI when we,blem 5 ofby choos-

o Options

hen a fea-

nl z-yalue

lor the LP

;asible in-

we solved

Group B

I , Suppose -we

have branched on a subproblem (call itsybqrob.le3 0,-having optimal solution SOLo) ar;;;v;obtained the following two subproblems:

$ublr0bhm I Subproblem 0 + ConstEinr rr = i.Sdpmbl€m 2. Subproblem 0 + Constrainr rr =, + I (i issome integer).

p,9* l!"t q:." will exist at least one optimat solution to5gbproblem I havingxl = i and at least on; optimal solution. .i?t"ql:T 2 having 11_ - i 1 1. lHint: Suppo; ;;opumat sotution to subproblem I lcall it SOLI) iris xl _;r. where.rr< i. For some numberc( 0 < c < I)..(SOL0)+ (l C)SOLI will have the following three properties: '

a The vatue of .rr in C(SOL0) + 1t .c)SOLt willequal i.

i _c(SOL0) + (l - c)SOLI will be feasible in sub_problem l.c The z-lalue for c{SOL0) r (l - C)SOLI will be atleast as good as the z-value for SOLI.

Explain how this result can help when we graphically solvelnoch-and-boutrd problems.l

During the next five periods, the demands h Table 5g* be met on time. At the beginning of period l, the

inventory level is 0. Each period tial production occurs asetup cost or'9250 and a per-unit production cost ofg2 arern-cu.rred. At the end of each period a per_unir holding costof $1 is incurred.

a . Solve for the cost-minimizing production scheduleusrng tle lbllowing decisioo variables: _r, = unil.s pro_

1f:9 9*", mon*r r and J.,, = I if -y *trs *" iio-ouced dunng period ,. /, -_ 0 otherwise.b

. Solve -for

the cost-minimiziDg produclion scheduleusng Oe following variables: y,.s difined in part 1a.1 andx" = number_of unils produced during period i to sat_lsty penod , demand.c Wlich formulation took LINDO or LINGO less timeto solve?

1..,9i:. I tnyu.": "xptanarion

of why rhe part (b) for-mutatton is solved faster than lhe pan (a) formuiaiion.

= 35,on

ICarl

523

TABLE 58

e.4 TIB Branch-and-Eound llrlethod lor $olvingl4ixed lnleger programming pmblems

Recall that, in a mixed If, some variables are required to be integers and others are al_lowed to be either integers or nonintegers. To solve urnk"a n Ly-tf," Uranch_and-boundmethod, modig, the method describetin Section 9.3 Uy lr_"iir! ".,V

., variables thatare required to be integers. Also, for a solution," "

,rup."ir"- ?" be a candidate solu-tion,.it need only assign integer values to.tf,o""_uUulf"r'tfiul'ar" .lq";r"0," be integers.To illustrate, Ier us solve the following mixed Ip:

maxz:2xt + x.z

s-t. 5x1 )_ 2x2 < g

x1 * x233x1, .r2 2 0; x1 integer

As before, we begin by solving the Lp reraxation of the Ip rhe optimar solution of theLP relaxal.ion is z : j.,, - i,*, = i. Because x, is ajlowed to ul nactlona, wedo nolbranch on x2; if we did so, we would b'e excluding-poil;".d;lr"s between 2 and3, and we don't want to do that. Thus, we must branch on.r1. This yields subproblems 2and 3 in Figure 21.We, next choose to solve subproblem 2. The optimal solution to subproblem 2 is thecandidate sorution z : 3. x1 - 0, x2 - 3. we now solve suuproir". i-a ouhin the can_didale sorution z : 1,,, = t. x, = ).me,-varr; ;;il';l;;;irJ,n r canoraare e,._ceeds the z-vatue for-the subprobiem i c-aiaut", .o ,uip.o ",ii ""#'i" "ri.irated

from::::id:*.,j-, and-the subprobrem 3 caadidate (" = I,i, :'t,.r:;iis the optimal so-ruhon to the mixed Ip

I .4 ne 0mllclrrd.0ol|d tteh0d l0r $0hiry Llired lnbpr fmgraln|llir0 pmthlns

In the cheapest-insertion heuristic (CIH), we begin at any city andneighbor. Then we create a subtour joining those two cities. Next, ',a,6

the subtour [say, arc (r,l)] by the combination of two arcs-1i, f) and (not in the cwrent subtour-that will. increase the lengh ofthe subtour bvcheapest) amount. We continue with this procedure until a tour ising this procedure begiruring with each city, we take the best tour found.

lmplicit Enumeration

In a 0-l IP, implicit enumeration may be used to find an optimal solution.ing at a node, create two new subproblems by (for some free variable x)straints .r, : 0 and xi : l. If the best completion of a node is feasible, thenbranch on the node. If the best completion is feasible and better than the

date solution, then the curent node yields a new LB (in a max problem) andtimal. If the best completion is feasible and is not better than the curenttion, then the curent node may be eliminated from consideration. Ifat a given

is at least one constaint that is not satisfied by any completion ofthe node,

cannot yield a feasible solution nor an optimal solution to the IP.

Cutting Plane Algorithm

step 1 Find the optimal tableau for the IP's linear programming relaxation. Ifables in the optimal solution assume integer values, we have found an optimalthe IP; otherwise, proceed to step 2.

Step 2 Pick a constraint in the LP relaxation optimal tableau whose righthandthe ftactional part closest to ]. This conshaint will be used to generate a cut.

Step 2a For the conshaint identified in step 2, wdte its right-hand side and each

ablet coefrcient in the form [r] +l where 0 ="f < 1.

Step 2b Rew te the constmint used to generate the cut as

All terms with integer coefficients : all terms with fractional coemcients

Then the cut is

All terms with iractional coefficients < 0

Slep 3 Use the clual simplex to find the optimal solution to the LP relaxation, with the

cut as an additional constraint. If all variables assume integer values in the optimal solu-

tion, then we have found an optimal solution to the IP Otherwise, pick the constraint witlt

the most fractional right-hancl side and use it to generate another cut, which is added to

the tableau. We continud this process until we obtain a solution in which all variables are

integers. This will be an optimal solution to the IP.

ecl&eeac

Fhe s,

m onro, lmintravel t

Cub

agent pSutt€

Tim S

the nulare s

the most vi

TABLE

I

2

3

4

5

6

TABLE

Dist ct

I2

3

4

5

RItIIEt]l, PROBLIll'lSGroup A1 In the Sailco problem of Section 3.10, suppose that afixed cost of $200 is incurred during each quarter thatproduction talies place. Formulate an IP to minimize Sailco'stotal cost of meeting the demands for the four quarters.

2 Explain how you would use inleger programming and

piecewise linear functions to solve the follo!4lntoptimization prcblem. (ffirr. Approximate f a114 y2 b!piecewise linear functions.)

lBased o

lBased o

clllPTfl 9 lllbger Pmumlllmillg

closest

1 alc inele /r iS

llest (orr apply-

L branch-

ing con-need notnt candi-

ay be op-

late solu-rde, there

r the node

If all vari-

solution to

rd side has

t each vari-

ients

on, with the

)Dtimal solu-

,nstraint with

;r is added to

vadabl€s ale

ma*z:3? + fs.t- x+ y=l

x,y>0

3- The Transyllania Olympic Cymnasrics Team consists

of six people. Transylvania must choose three people to

enter both the balarce beam and floor elrcrcises. They must

also enter a total offour people in each event. The score that

"66tr irdividual g]4nnasl can attain in each event is shown

in Table 89. Formulate an [P ro maximize rhe rotal score

aMined by the Transylvania gymnasts.

41 6 court decision has stated that the enrollment of eachgh school in Metropolis must be at least 20 percenr black.

fte numbers of black and white high school srudents ineaoh of the cityt five school districts are shown in Table 90.

The distance (in miles) that a student ir each distdct musttravel to each high school is shown in Table 91. School

66ud policy requires that all the students in a giver districtn69nd the same school. Assuming that each school must

have an enrollrnent ofat least 150 students, formulate ar IPlhat will minimize the total distance that Metropolis students

must havel to high school.

TABLE 9'tlish fiishDislrict School i School 2

I2

3

4

5

I0.5

0.81.3

1.5

2

t.'7

0.8

0.4

0.6

TABLE 92

RS

BS

DEST

TS

6

4

3

z2

6 (richty)

5 (dchty)

3 (righty)

3 (lefty)

2 (righty)

5 The Cubs are trying to determine which ofthe followingt€e agent pitchers should be signed: Rick Sutcliffe (RS),Bruce Suttn (BS), Dennis Eckersley (DE), Steve Trout

6T), Tim Stoddard (TS). The cost of signing each pitcherand the number of victories each pitcher will add to theoubs are shown in Table 92. Subject to the following

. Fstrictions, the Cubs want to sign the pitchers who will add

'.ile most victories to the team.

a At most, $12 million can be spent.

t If DE and ST arc signed, then BS canaot be signed.

c At most two right-handed pitchers can be signed.

d The Cubs carmot sign both BS and RS.

Formulate an IP to help the Cubs determine who they shouldslgll

6 State Unive6ity must puchase 1,100 computers fromthree vendors. Vendor I charyes $500 per computer plus adelivery charge of $5,000. Vendor 2 charyes $350 percomputer plus a delivery charge of $4,000. Vendor 3 chaiges$250 per computer plus a delivery charge of $6,000. VendorI will sell the university at most 500 computem; vendor 2,at most 900; and vendor 3, at most 400. Formulate aIl IP tominimize the cost ofpurchasing the needed computerc.

7 Use the branch-and-bound method to solve the follow-ing IP:

m?Lxz=3tr+x2s.t, 5xt + x2 = 12

2x1 * x238xr, x2 > 0; rt, x2 ir'teget

8 Use the branch-and-bound method to solve the follow-ing IP:

mitrz:3x1 *x2s.t. 2.xr - x2 = 6

xr+x2=4- xt, x2 > 0; ,r integer

9 Use the branch-and-bound method to solve the follow-irg IP:

maxz = xr + 2,a2

s.t. ,l + Jr2 = l02\+5x2<30

xt, )b > 0; xr, x2 i,,tqBet

BLE 89

8.8 7.9

9.4 8.3

9.2

7.5

8.1

9-1

8.5

8.7

8.1

8.6

80'70

90

50

60

30

5

l040

30

,,r#.ili1&q4

. rz and! "'

RsYie[ Pmblelns 553